| L(s) = 1 | + 2-s + 2.73·3-s + 4-s + 2.73·5-s + 2.73·6-s − 7-s + 8-s + 4.46·9-s + 2.73·10-s + 2.73·12-s + 4.73·13-s − 14-s + 7.46·15-s + 16-s − 6.92·17-s + 4.46·18-s + 0.732·19-s + 2.73·20-s − 2.73·21-s − 6.92·23-s + 2.73·24-s + 2.46·25-s + 4.73·26-s + 3.99·27-s − 28-s − 7.46·29-s + 7.46·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.57·3-s + 0.5·4-s + 1.22·5-s + 1.11·6-s − 0.377·7-s + 0.353·8-s + 1.48·9-s + 0.863·10-s + 0.788·12-s + 1.31·13-s − 0.267·14-s + 1.92·15-s + 0.250·16-s − 1.68·17-s + 1.05·18-s + 0.167·19-s + 0.610·20-s − 0.596·21-s − 1.44·23-s + 0.557·24-s + 0.492·25-s + 0.928·26-s + 0.769·27-s − 0.188·28-s − 1.38·29-s + 1.36·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.344275043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.344275043\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 19 | \( 1 - 0.732T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 7.46T + 29T^{2} \) |
| 31 | \( 1 - 0.535T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 0.732T + 61T^{2} \) |
| 67 | \( 1 - 9.46T + 67T^{2} \) |
| 71 | \( 1 - 7.46T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.428552389793058544842305283334, −8.529906328235592364184431836232, −7.956239136068623248769392087973, −6.68625549008324176156411314107, −6.27812793800013075803701711864, −5.22742605487131367020361527896, −3.99853758733047675536312557993, −3.43721334514567236671051619268, −2.25802704295686077769074416849, −1.82832118518285429620608730283,
1.82832118518285429620608730283, 2.25802704295686077769074416849, 3.43721334514567236671051619268, 3.99853758733047675536312557993, 5.22742605487131367020361527896, 6.27812793800013075803701711864, 6.68625549008324176156411314107, 7.956239136068623248769392087973, 8.529906328235592364184431836232, 9.428552389793058544842305283334
